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Arithmetic of Logic* ARITHMETIC OF LOGIC* BY E. T. BELL 1. Introduction. This is probably the first attempt to construct an arith- metic for an algebra of the non-numerical genus.f In his classic treatise, An Investigation of the Laws of Thought,J Boole developed the thesis that "Logic (is) • • • a system of processes carried on by the aid of symbols having a definite interpretation, and subject to laws founded on that interpretation alone. But at the same time they exhibit those laws as identical in form with the laws of the general symbols of Algebra, with this simple addition, viz., that the symbols of Logic are further subject to a special law, to which the symbols of quantity as such, are not subject." The special law is what Boole calls the law of duality, x(l—x)=0, or the excluded middle; here the indi- cated multiplication is logical, 1—a: is the supplement of x. Boole showed therefore that abstractly logic is contained in common algebra. Taking rational arithmetic Si ( = the theory of numbers in reference to the positive rational integers 0, 1, 2, 3, • • • , only) and certain parts of the theory of algebraic numbers, particularly the rudiments of Dedekind's theory of ideals and those of Kronecker's modular systems as our guides, we shall see to what extent Boole's algebra of logic may be arithmetized in a precise sense to be defined presently. Although it will be unnecessary to refer explicitly anywhere to the theory of algebraic numbers, it may be mentioned that this theory, which includes rational arithmetic, is a surer guide than the latter in problems of arithmetization. In rational arithmetic the essential abstract structure of the concepts to be extended beyond 0, 1, 2, • • • is often quite ingeniously concealed. This is true, for example, of the G.C.D., L.C.M., and residuation. The theory of ideals, on the other hand, often indicates immediately what transformations by formal equivalence must first be applied to operations or relations of rational arithmetic in order that they shall be significant for sets of elements for which order relations are either irrelevant or meaningless. * Presented to the Society, San Francisco Section, April 2, 1927; received by the editors Novem- ber 29, 1926. t For the meaning of this term, cf. Whitehead, Universal Algebra, p. 29. X London and Cambridge, 1854; reprinted, Chicago, Open Court, 1916, as vol. 2 of Boole's CollectedLogical Works. 597 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 598 E. T. BELL July We shall use a few of the commonest notations of algebraic logic ; thus PdQ for P implies Q; P: = :Q for P, Q, are formally equivalent, viz., Pz>Q. QdP, the dot in the last being the logical and; = signifies definitional identity, except where it occurs in conjunction with mod, when it indicates congruence, as in a=ß mod n; the special notation a\ß, borrowed from the theory of numbers, where a, ß are classes, signifies a contains ß, = each element of ß is in a. To avoid confusion we shall never use "contains" in relation to arith- metic; conflicting conventions in this respect have already introduced exasperating paradoxes of language into the theory of numbers. Small Greek letters a, ß, 7, • • • will always denote classes ; S is the set of classes discussed; the null class is denoted by a, the universal class by e, so that w, e are the zero, unity of the algebra of logic 8. Elements of © will be called elements of 8. Logical (but not arithmetical) addition, multiplication of classes a, ß are indicated as usual by a+ß, aß, and if a is any element of 8, the supplement of a is indicated by an accent, a' (instead of the customary bar which is awkward in monotype). Hence a' is the unique* solution of a-f-a' = e, aa' = w, where a is any given element of 8. We shall assume a given set 6 of classes and we postulate that if a, ß are any elements of (£, then a', a+ß, aß are in (5, and, as before, we refer to elements of S as ele- ments of 8. The letters s, p, l, g denote specific operations upon elements of 8, and they are such that (once for all, without further reference) atß for each of t = s,P, l, g is a uniquely determined element of 8 ; the letters c, d, r denote specific relations such that a t ß for each t = c, d, r is uniquely significant in 8. To assist the memory we remark that s, p, l, g, c, d, r may be read sum, product, least, greatest, congruent, divides, residual, terms to be defined in the arithmetic of logic as opposed to the algebra. The least, greatest here are intended merely to recall classes having with respect to given classes properties abstractly identical with those of the G.C.D., L.C.M. with respect to division in 21; these classes g, I are not necessarily the "most" or the "least" inclusive—the rôles with respect to inclusion may be reversed, and g may be either the most or the least inclusive common class of a set, and similarly for I. Hence we shall not use here the names already familiar in Moore's general analysis. The zero, unity in the arithmetic of classes will always be denoted by f, v. It will avoid possible confusion if we add that (f, v) are not necessarily equal to (a, e) respectively. * For proof of unicity cf. Whitehead, Universal Algebra, p. 36. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1927] ARITHMETIC OF LOGIC 599 Suppose all the elements, operations and signs of relations, other than the logical constants, are replaced in a set SEof propositions by marks without significance beyond that implied by the assertion of the propositions (which include the postulates of the set). Call the result, C(t), the content of SE. If in C(SE) it be possible to assign interpretations to the marks, giving SE,-, such that SE,-is self-consistent and uniquely significant in terms of the in- terpretation, we shall call SE,-an instance of C(SE). Sets X, (j = l, 2, ■ ■ • ) of propositions having the same content will be called abstractly identical. Our object is to find parts 21,of rational arithmetic 21abstractly identical with parts ?,- of the algebra of classes, hence also with the algebra of relations, and finally with 8. In such a project the following type of invariance under formal equivalence is extremely useful. Let P,- (j = l, 2, ■ ■ • ) be propositions of SEsuch that Pi: ■ :P2. Then the truth-values of Pi oPz, Pz sPi are identical with those of P2 3 Pz, Pz ^Ps- Thus : = : in propositions is abstractly identical with = in common algebra, and in implications a particular proposition may be replaced by any other which is formally equivalent to it. We shall meet several instances of such transformations which are considerably less obvious than the logic which justifies them. The identical transformation of a set of transformations by formal equivalence is that which replaces each proposition of a given set by itself; any set of transformed propositions (including the set transformed by the identical transformation) is called a transform of the original set. Let SE' be a transform of SE,and 21/ a transform of a part 21,-of 21. Then if SE', 21/ are abstractly identical, we shall say that SE is arithmetized with respect to 21,, or simply arithmetized. This kind of arithmetization can be carried much farther for 8 than is done here, but what is given will suffice to show its nature, and it will be evident that at nearly every stage there are alternative ways of proceeding. We shall exhibit arithmetizations of 8 with respect to congruences, the L.C.M., G.C.D., divisibility, primes, and the unique factorization law. Rational arithmetic 21 presupposes the existence of a special ring. In the whole discussion we shall ignore negative numbers, without loss of generality, as the arithmetic (properties of integers as such) which refers to these can always be thrown back to relations between positive integers only, e.g., as done by Kronecker. An abstract ring 3Í is a set © of elements x, y, z, ■ ■ • , u', z', ■ ■ • , and two operations S, P( = addition, multiplication) which may be performed upon any two equal or distinct elements x, y of 3î, in this order, to produce uniquely determined elements xSy, xPy such that the postulates 9Î,-(j=1,2, 3) are satisfied. Elements of © will be called elements of 9Î. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 600 E. T. BELL July 9îi. If x, y are any two elements of 3Î, xSy, xPy are uniquely determined elements of 9î, and ySx = xSy, yPx = xPy. 9Î2. If x, y, z are any three elements of 5R, (xSy)Sz = xS(ySz), (xPy)Pz = xP(yPz), xP(ySz) = (xPy)S(xPz). 5R3. There exist in 3Î two distinct* unique elements, denoted by «', z', and called the unity, zero of 9Î, such that if x is any element of 9Î, xSz' = x, xPu' = x. These may be compared with the first three postulates of Dicksont for a field, of which they are a transcription, except that the unicity of «', z' is here a postulate, not a theorem, also with Wedderburn'sJ for algebraic fields. In each comparison the omissions are to be particularly noticed.
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